United took the lead on 7 minutes through American Kenny Cooper, but created little else as Williams performed wonders to keep City at bay.
The United keeper prevented both Christian Negouai and Stephen Elliott from scoring the equaliser.
But the leveller eventually came on 41 minutes through Bradley Wright-Phillips.
Negoaui then had an effort cleared off the line just after the break while Wright-Phillips was also denied by Williams.
City finally got the winner they deserved on 76 minutes thorugh Irish striker, Stephen Elliott.
Lee Croft's corner was cleared but headed back in by Stephen Jordan. Elliott then tuned to fire in the winner.
City: Schmeichel, Flood, Jordan, Wiekens, McCarthy, Negouai (Tandy 69), Croft, J D'Laryea, Tiatto (N D'Laryea 85), Elliott, B Wright-Phillips,
Subs: Matthewson(GK), Onouha, Murphy.
United: B Williams, Sims (Lawrence 69), Pugh, Bardsley, Tate, Wood, M Williams, Johnson (Timm 69), Cooper (Poole 60), Nardiello, Richardson
Subs: Jowsey(GK), Lynch.
Attendance:2,197
At Ewen Fields, Hyde United FC